Just as well. The OP's post was worded like a homework assignment. Think about it. He says he "has a transducer.." . He doesn't say "I have THIS transducer [link] . He doesn't say WHAT he doing. He doesn't say WHY he wants ringing. He says he wants to use an op amp but obviously does not know ANYTHING about op amps. Does not even know what a Voltage Follower is because he describes one to a T but not by name. He describes an LPF but not by name. Instead of saying " Can I use an op amp to generate such and such an output from a [ sensor name] input ?" he starts out by asking "what op amp configuration follows the voltage of it's input ?" (is that a trick question?)Obviously he's never used an op amp.. He asks for an " under damped controlled system but doesn't know the first thing about Control Systems (like PID parameters for starters).
We have more free parameters than we really need so we can set R1=R2=R.
I have a transducer that creates a step voltage (0 to 4.5V) with a proximity stimulus, However I would like to put this step input into an op amp where the output of the op amp will follow the step input but with a controlled amount of ringing (both amplitude and settling time). This is essentially a controlled under-damped system.